The present invention relates to a digital filter apparatus having a resonance characteristic.
As an alternative to an analog filter incorporating a transistor, a resistor, a capacitor, a coil and an operational amplifier, digital filters incorporating a digital circuit such as a multiplier, an adder, or a delay circuit have recently received much attention. Such a digital filter is used, for example, as a tone color setting circuit in an electronic musical instrument. These digital filters may be, for example, low-pass filters, high-pass filters and band-pass filters. For example, some analog type music synthesizers incorporate an analog filter having a resonance characteristic in order to add special tone color to the sound. When an analog filter having a resonance characteristic is used, a peak is generated in the amplitude of the musical sound signal and a musical sound is obtained which has a special tone color in which this frequency component having the peak is emphasized. Although it is possible to construct a digital filter having such a resonance characteristic, a ROM of large capacity as an element of the digital filter is generally necessary, resulting in a disadvantage. This will be described in more detail.
For designing a filter, a transfer function must first be obtained. A method is known for designing a digital filter according to which a transfer function H(S) of an analog filter is first obtained, and then it undergoes the standard Z-transform, the bilinear Z-transform, or the alignment Z-transform to obtain a transfer function H(Z) of the desired digital filter.
An example of an analog low-pass filter of a second-order will be described. A transfer function H(S) of the second-order low-pass filter is generally expressed by ##EQU1##
FIG. 1 shows the unit circle on the S-plane and the poles as represented by equation (1). When Q=1, the poles may be expressed as ##EQU2## which are indicated by P1, P2 in the figure. In the equation (1), Q is the amplitude of resonance and is usually 1 under the normal non-resonance condition.
For realizing a resonance characteristic, the poles are moved along the unit circle to the imaginary axis as indicated by arrows in FIG. 1. In this case, a peak is generated in the amplitude characteristic at the position .omega.=.omega.0 (resonance angular frequency) as shown in FIG. 2.
A digital filter apparatus having such a resonance characteristic has been described in a U.S. patent application (Ser. No. 279,630, filed on July 1, 1981). In this application, the apparatus can be realized by merely adding relatively simple circuitry having a memory of small capacity. In the respective embodiments described in this application, the resonance characteristic has been realized by moving the poles on the Z-plane parallel to the imaginary axis as shown in FIG. 3 to approach the unit circle. However, when the poles are moved parallel to the imaginary axis, the cutoff frequency is increased as the amplitude of resonance becomes larger (or decreased when the poles are in the left half-plane). FIG. 4 shows this case, wherein the cutoff frequency is plotted as the abscissa and the gain is plotted as the ordinate. Therefore, there are cases where it is difficult to obtain the resonance characteristic at the desired value of the cutoff frequency. Further, for achieving the resonance characteristic in the low frequency region, a considerably larger number of operation bits is required.